This paper presents a general probabilistic model for transcribing single-channel music recordings containing multiple polyphonic instrument sources. The system requires no prior knowledge of the instruments present in the mixture (other than the number), although it can benefit from information about instrument type if available. In contrast to many existing polyphonic transcription systems, our approach explicitly models the individual instruments and is thereby able to assign detected notes to their respective sources. We use training instruments to learn a set of linear manifolds in model parameter space which are then used during transcription to constrain the properties of models fit to the target mixture. This leads to a hierarchical mixture-of-subspaces design which makes it possible to supply the system with prior knowledge at different levels of abstraction. The proposed technique is evaluated on both recorded and synthesized mixtures containing two, three, four, and five instruments each. We compare our approach in terms of transcription with (i.e., detected pitches must be associated with the correct instrument) and without source-assignment to another multi-instrument transcription system as well as a baseline non-negative matrix factorization (NMF) algorithm. For two-instrument mixtures evaluated with source-assignment, we obtain average frame-level F-measures of up to 0.52 in the completely blind transcription setting (i.e., no prior knowledge of the instruments in the mixture) and up to 0.67 if we assume knowledge of the basic instrument types. For transcription without source assignment, these numbers rise to 0.76 and 0.83, respectively.